Study of γγ → γψ ð 2S Þ at Belle
X. L. Wang ,11B. S. Gao,11W. J. Zhu,11I. Adachi,19,15H. Aihara,91S. Al Said,84,39D. M. Asner,3 H. Atmacan,7 V. Aulchenko,4,68T. Aushev,21R. Ayad,84V. Babu,8S. Bahinipati,25P. Behera,28V. Bhardwaj,24B. Bhuyan,26T. Bilka,5 J. Biswal,35A. Bobrov,4,68G. Bonvicini,94A. Bozek,64M. Bračko,51,35M. Campajola,33,59D.Červenkov,5M.-C. Chang,10 V. Chekelian,52A. Chen,61B. G. Cheon,17K. Chilikin,46H. E. Cho,17K. Cho,41S.-K. Choi,16Y. Choi,82S. Choudhury,27
D. Cinabro,94S. Cunliffe,8 S. Das,50G. De Nardo,33,59 R. Dhamija,27F. Di Capua,33,59 Z. Doležal,5 T. V. Dong,11 S. Eidelman,4,68,46T. Ferber,8 D. Ferlewicz,53A. Frey,14B. G. Fulsom,70 R. Garg,71V. Gaur,93N. Gabyshev,4,68 A. Garmash,4,68A. Giri,27P. Goldenzweig,36B. Golob,47,35C. Hadjivasiliou,70T. Hara,19,15O. Hartbrich,18K. Hayasaka,66
H. Hayashii,60M. T. Hedges,18W.-S. Hou,63C.-L. Hsu,83T. Iijima,58,57K. Inami,57A. Ishikawa,19,15 R. Itoh,19,15 M. Iwasaki,69Y. Iwasaki,19W. W. Jacobs,29S. Jia,11 Y. Jin,91C. W. Joo,37K. K. Joo,6J. Kahn,36K. H. Kang,44 T. Kawasaki,40C. Kiesling,52C. H. Kim,17D. Y. Kim,81S. H. Kim,78Y.-K. Kim,96P. Kodyš,5T. Konno,40A. Korobov,4,68
S. Korpar,51,35 E. Kovalenko,4,68P. Križan,47,35 R. Kroeger,54P. Krokovny,4,68 R. Kulasiri,38M. Kumar,50R. Kumar,74 K. Kumara,94A. Kuzmin,4,68Y.-J. Kwon,96K. Lalwani,50J. S. Lange,12I. S. Lee,17S. C. Lee,44P. Lewis,2J. Li,44L. K. Li,7
Y. B. Li,72L. Li Gioi,52J. Libby,28K. Lieret,48D. Liventsev,94,19C. MacQueen,53 M. Masuda,90,75 T. Matsuda,55 D. Matvienko,4,68,46M. Merola,33,59 F. Metzner,36K. Miyabayashi,60R. Mizuk,46,21G. B. Mohanty,85M. Mrvar,32
R. Mussa,34M. Nakao,19,15 Z. Natkaniec,64 A. Natochii,18L. Nayak,27M. Nayak,87M. Niiyama,43N. K. Nisar,3 S. Nishida,19,15K. Nishimura,18S. Ogawa,88H. Ono,65,66Y. Onuki,91P. Oskin,46P. Pakhlov,46,56G. Pakhlova,21,46T. Pang,73 S. Pardi,33H. Park,44S.-H. Park,19S. Patra,24S. Paul,86,52T. K. Pedlar,49R. Pestotnik,35L. E. Piilonen,93T. Podobnik,47,35 V. Popov,21E. Prencipe,22M. T. Prim,2M. Röhrken,8A. Rostomyan,8N. Rout,28G. Russo,59D. Sahoo,85S. Sandilya,27 A. Sangal,7L. Santelj,47,35T. Sanuki,89G. Schnell,1,23C. Schwanda,32Y. Seino,66K. Senyo,95M. E. Sevior,53M. Shapkin,31 C. Sharma,50C. P. Shen,11J.-G. Shiu,63B. Shwartz,4,68F. Simon,52J. B. Singh,71A. Sokolov,31E. Solovieva,46S. Stanič,67 M. Starič,35Z. S. Stottler,93M. Sumihama,13M. Takizawa,79,20,76U. Tamponi,34F. Tenchini,8M. Uchida,92S. Uehara,19,15 T. Uglov,46,21Y. Unno,17S. Uno,19,15P. Urquijo,53Y. Usov,4,68R. Van Tonder,2 G. Varner,18A. Vossen,9 E. Waheed,19 C. H. Wang,62M.-Z. Wang,63P. Wang,30M. Watanabe,66S. Watanuki,45O. Werbycka,64E. Won,42X. Xu,80W. Yan,77
S. B. Yang,42H. Ye,8 J. H. Yin,42C. Z. Yuan,30Z. P. Zhang,77V. Zhilich,4,68 and V. Zhukova46 (Belle Collaboration)
1Department of Physics, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain
2University of Bonn, 53115 Bonn, Germany
3Brookhaven National Laboratory, Upton, New York 11973, USA
4Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russian Federation
5Faculty of Mathematics and Physics, Charles University, 121 16 Prague, The Czech Republic
6Chonnam National University, Gwangju 61186, South Korea
7University of Cincinnati, Cincinnati, Ohio 45221, USA
8Deutsches Elektronen–Synchrotron, 22607 Hamburg, Germany
9Duke University, Durham, North Carolina 27708, USA
10Department of Physics, Fu Jen Catholic University, Taipei 24205, Taiwan
11Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute of Modern Physics, Fudan University, Shanghai 200443, People’s Republic of China
12Justus-Liebig-Universität Gießen, 35392 Gießen, Germany
13Gifu University, Gifu 501-1193, Japan
14II. Physikalisches Institut, Georg-August-Universität Göttingen, 37073 Göttingen, Germany
15SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193, Japan
16Gyeongsang National University, Jinju 52828, South Korea
17Department of Physics and Institute of Natural Sciences, Hanyang University, Seoul 04763, South Korea
18University of Hawaii, Honolulu, Hawaii 96822, USA
19High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan
20J-PARC Branch, KEK Theory Center, High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan
21Higher School of Economics (HSE), Moscow 101000, Russian Federation
22Forschungszentrum Jülich, 52425 Jülich, Germany
23IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain
24Indian Institute of Science Education and Research Mohali, Sahibzada Ajit Singh Nagar, 140306, India
25Indian Institute of Technology Bhubaneswar, Bhubaneswar 752050, India
26Indian Institute of Technology Guwahati, Assam 781039, India
27Indian Institute of Technology Hyderabad, Telangana 502285, India
28Indian Institute of Technology Madras, Chennai 600036, India
29Indiana University, Bloomington, Indiana 47408, USA
30Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
31Institute for High Energy Physics, Protvino 142281, Russian Federation
32Institute of High Energy Physics, Vienna 1050, Austria
33INFN—Sezione di Napoli, 80126 Napoli, Italy
34INFN—Sezione di Torino, 10125 Torino, Italy
35J. Stefan Institute, 1000 Ljubljana, Slovenia
36Institut für Experimentelle Teilchenphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe, Germany
37Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa 277-8583, Japan
38Kennesaw State University, Kennesaw, Georgia 30144, USA
39Department of Physics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
40Kitasato University, Sagamihara 252-0373, Japan
41Korea Institute of Science and Technology Information, Daejeon 34141, South Korea
42Korea University, Seoul 02841, South Korea
43Kyoto Sangyo University, Kyoto 603-8555, Japan
44Kyungpook National University, Daegu 41566, South Korea
45Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
46P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow 119991, Russian Federation
47Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia
48Ludwig Maximilians University, 80539 Munich, Germany
49Luther College, Decorah, Iowa 52101, USA
50Malaviya National Institute of Technology Jaipur, Jaipur 302017, India
51Faculty of Chemistry and Chemical Engineering, University of Maribor, 2000 Maribor, Slovenia
52Max-Planck-Institut für Physik, 80805 München, Germany
53School of Physics, University of Melbourne, Victoria 3010, Australia
54University of Mississippi, University, Mississippi 38677, USA
55University of Miyazaki, Miyazaki 889-2192, Japan
56Moscow Physical Engineering Institute, Moscow 115409, Russian Federation
57Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan
58Kobayashi-Maskawa Institute, Nagoya University, Nagoya 464-8602, Japan
59Universit `a di Napoli Federico II, 80126 Napoli, Italy
60Nara Women’s University, Nara 630-8506, Japan
61National Central University, Chung-li 32054, Taiwan
62National United University, Miao Li 36003, Taiwan
63Department of Physics, National Taiwan University, Taipei 10617, Taiwan
64H. Niewodniczanski Institute of Nuclear Physics, Krakow 31-342, Poland
65Nippon Dental University, Niigata 951-8580, Japan
66Niigata University, Niigata 950-2181, Japan
67University of Nova Gorica, 5000 Nova Gorica, Slovenia
68Novosibirsk State University, Novosibirsk 630090, Russian Federation
69Osaka City University, Osaka 558-8585, Japan
70Pacific Northwest National Laboratory, Richland, Washington 99352, USA
71Panjab University, Chandigarh 160014, India
72Peking University, Beijing 100871, People’s Republic of China
73University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
74Punjab Agricultural University, Ludhiana 141004, India
75Research Center for Nuclear Physics, Osaka University, Osaka 567-0047, Japan
76Meson Science Laboratory, Cluster for Pioneering Research, RIKEN, Saitama 351-0198, Japan
77Department of Modern Physics and State Key Laboratory of Particle Detection and Electronics, University of Science and Technology of China, Hefei 230026, People’s Republic of China
78Seoul National University, Seoul 08826, South Korea
79Showa Pharmaceutical University, Tokyo 194-8543, Japan
80Soochow University, Suzhou 215006, China
81Soongsil University, Seoul 06978, South Korea
82Sungkyunkwan University, Suwon 16419, South Korea
83School of Physics, University of Sydney, New South Wales 2006, Australia
84Department of Physics, Faculty of Science, University of Tabuk, Tabuk 71451, Saudi Arabia
85Tata Institute of Fundamental Research, Mumbai 400005, India
86Department of Physics, Technische Universität München, 85748 Garching, Germany
87School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
88Toho University, Funabashi 274-8510, Japan
89Department of Physics, Tohoku University, Sendai 980-8578, Japan
90Earthquake Research Institute, University of Tokyo, Tokyo 113-0032, Japan
91Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
92Tokyo Institute of Technology, Tokyo 152-8550, Japan
93Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA
94Wayne State University, Detroit, Michigan 48202, USA
95Yamagata University, Yamagata 990-8560, Japan
96Yonsei University, Seoul 03722, South Korea
(Received 6 January 2022; accepted 15 June 2022; published 28 June 2022)
Using980fb−1of data at and around theϒðnSÞ(n¼1, 2, 3, 4, 5) resonances collected with the Belle detector at the KEKB asymmetric-energyeþe−collider, the two-photon processγγ→γψð2SÞis studied from the threshold to 4.2 GeV for the first time. Two structures are seen in the invariant mass distribution of γψð2SÞ: one atMR1 ¼3922.46.52.0MeV=c2with a width ofΓR1¼22174MeV, and another at MR2¼4014.34.01.5MeV=c2 with a width of ΓR2¼4116MeV; the signals are para- metrized with the incoherent sum of two Breit-Wigner functions. The first structure is consistent with the Xð3915Þor theχc2ð3930Þ, and the local statistical significance is determined to be3.1σwith the systematic uncertainties included. The second matches none of the known charmonium or charmoniumlike states, and its global significance is determined to be2.8σ including the look-elsewhere effect. The production rates areΓγγBðR1→γψð2SÞÞ ¼9.83.61.3eV assumingðJPC;jλjÞ ¼ ð0þþ;0Þor2.00.70.2eV withð2þþ;2Þfor the first structure andΓγγBðR2→γψð2SÞÞ ¼6.22.20.8eV withð0þþ;0Þor1.2 0.40.2eV withð2þþ;2Þfor the second. Here, the first errors are statistical and the second systematic, and λis the helicity.
DOI:10.1103/PhysRevD.105.112011
I. INTRODUCTION
More than two dozen new resonances that are dubbed asX, Yand/orZstates have been found above theDD¯ threshold since Belle observed the Xð3872Þ (now labeled the χc1ð3872Þ[1]) inB→Kπþπ−J=ψ [2], and this number is much larger than the expectation from predictions of the conventional model. Among these, candidates for both conventional and exotic charmoniumlike states are discussed widely[3]. Many puzzles arise from theseXYZstates, and one of them concerns the candidates forP-wave triplet states near 3.9 GeV=c2, including theXð3872Þ,Zð3930Þ→DD¯ and Xð3915Þ→ωJ=ψ observed in two-photon collisions [4–8], and Xð3860Þ→DD¯ observed in a full amplitude analysis of the processeþe−→J=ψDD¯ [9].
One of the most interestingXYZ states is theXð3872Þ, which lies very near theDD¯þc:c:mass threshold and is conjectured to have a largeDD¯þc:c:molecular compo- nent[10]. Its large production rates inppandpp¯ collision experiments[11–14]and the determination of its quantum
number by LHCb[15]suggest that there is a conventional charmonium χc1ð2PÞ core in its wave function. This is supported by another study ofXð3872Þ→γψð2SÞby LHCb [16]. A study of the line shape of this state by LHCb reveals a pole structure that is compatible with a quasibound state of D0D¯0but allowing a quasivirtual state at the level of2σ[17].
Partners of theXð3872Þare suggested, and one of them is a DD¯ loosely bound state with quantum numbers JPC¼ 2þþ[18,19]. The recent study of the production cross section of theXð3872Þ relative to the ψð2SÞ in ppcollisions by LHCb shows that theXð3872Þproduction is less suppressed relative to the promptψð2SÞin the higherpTregion [20], which is similar to the case ofψð2SÞrelative toJ=ψ[21,22].
Belle found evidence forXð3872Þproduction in two-photon collisions[23], thus motivating the search for the possible 2þþpartner of theXð3872Þin such collisions. Such a study can provide essential information to understand the nature of theXð3872Þ.
Concurrently, there have been many studies related to theχcJð2PÞtriplet states. TheZð3930Þwas discovered by
Belle in the process γγ→DD¯, and the angular distribu- tion was used to identify it as the χc2ð2PÞ state [4]. The existence of Zð3930Þ and its angular distribution were confirmed byBABAR[5]. TheXð3915Þwas discovered by Belle[7]and a spin-parity analysis of this state byBABAR favored theJPC¼0þþquantum numbers[8]. TheXð3915Þ is a candidate of the χc0ð2PÞ state [24–27]. In a recent amplitude analysis of theBþ→KþDþD−decay by LHCb [28], there are both 0þþ and 2þþ states at mðDþD−Þ≈
3930MeV=c2. Their parameters are determined to be M¼3923.81.50.4MeV=c2 and Γ¼17.45.1 0.8MeV for χc0ð3930Þ and M¼3926.82.4 0.8MeV=c2 andΓ¼34.26.61.1MeV forχc2ð3930Þ.
(Here and hereafter, the first errors are statistical and the second are systematic.) The χc2ð3930Þ state is a good candidate of χc2ð2PÞ, but this would imply that the hyperfine splitting of 12MeV=c2 between χc2ð2PÞ and Xð3915Þ would be only 6% of that between χc2ð1PÞ and χc0ð1PÞ[29]. In contrast, an early calculation[30]utilizing the Godfrey-Isgur relativistic potential model[31]predicts a much larger mass difference of about60MeV=c2 [30].
The Xð3860Þ observed by Belle is another candidate of χc0ð2PÞ but it was not seen in LHCb’s study of the Bþ →KþDþD− decay [28]. One interpretation of the Xð3860Þis aDD¯ bound state close to the threshold with isospin I¼0 [32]. Therefore, additional studies of the P-wave triplet states near 3.9 GeV=c2 are needed for a more comprehensive understanding of theXYZstates and, in particular, of the Xð3872Þ.
Both0þþand2þþstates can be produced in two-photon collisions and can decay toγψð2SÞvia an E1 transition. For example, the partial widths are expected to beΓðχc0ð2PÞ→ γψð2SÞÞ≈135keV andΓðχc2ð2PÞ→γψð2SÞÞ≈207keV according to the aforementioned calculation [30]. In this article, we report an investigation of theγψð2SÞfinal state produced in two-photon collisions [eþe− →eþe−γγ→ eþe−γψð2SÞ or γγ →γψð2SÞ for brevity], using data collected with the Belle detector [33] at the KEKB asymmetric-energy eþe− collider [34]. The ψð2SÞ is reconstructed from its hadronic final stateπþπ−J=ψ with J=ψ reconstructed from a lepton pairlþl−ðl¼e;μÞ.
II. DETECTOR, DATA SAMPLE, AND MONTE CARLO (MC) SIMULATION The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector, a 50-layer central drift chamber, an array of aerogel threshold Cherenkov counters, a barrel-like arrangement of time-of- flight scintillation counters, and an electromagnetic calo- rimeter (ECL) comprised of CsI(Tl) crystals located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux return located outside of the coil is instrumented to detect K0L mesons and to identify muons. The origin of the coordinate system is defined as
the position of the nominal interaction point (IP). Thezaxis is aligned with the direction opposite theeþ beam and is parallel to the direction of the magnetic field within the solenoid. The x axis is horizontal and points toward the outside of the storage ring; they axis is vertical upward.
The polar angle θ and azimuthal angle ϕ are measured relative to the positivez andxaxes, respectively.
The integrated luminosity of Belle data used in this analysis is980fb−1. About 70% of the data are collected at theϒð4SÞresonance, and the rest are taken at otherϒðnSÞ (n¼1, 2, 3, or 5) states or center-of-mass (c.m.) energies a few tens of MeV below the ϒ states. The TREPS event generator[35]is used to simulate the signals ofγγ →X → γψð2SÞ for optimization of selection criteria, efficiency determination and calculation of the luminosity function Lγγ of two-photon collisions in Belle data. Here, X is χc2ð3930Þ, Xð3915Þ or a resonance with mass fixed to a value between 3.8 and4.2GeV=c2and width fixed to zero.
In the production ofγγ →X, the helicityλis the direction of theγγaxis in the rest frame ofX. Thejλj ¼2component is reported to dominate in the measurements of γγ → χc2ð3930Þ→DD¯ by Belle [4]andBABAR[5]. A sample ofχc2ð3930Þwith helicityjλj ¼2is taken to be the nominal signal MC sample. The major background is found to be the initial-state radiation (ISR) process eþe−→ψð2SÞ, which has a cross section of 15.420.120.89fb in the Belle data sample[36]. There are0.6×106events with a πþπ−lþl− final state in data, and an MC sample containing 3.8×106 such events is simulated with the
PHOKHARA generator, which has a precision better than 0.5%[37]. An MC simulation usingGEANT3[38]is used to model the performance of the Belle detector.
III. SELECTION CRITERIA AND SIGNAL RECONSTRUCTIONS
Photon candidates are reconstructed from ECL clusters that do not match any charged tracks; the candidate with the highest energy is selected to form theγψð2SÞfinal state. This energy is required to be larger than 100 MeV to suppress the background from fake photons. A candidate of ψð2SÞ→ πþπ−J=ψ withJ=ψ →eþe−orμþμ−is reconstructed from four well-measured charged tracks, each having impact parameters with respect to the IP ofjdzj<5cm along the z(positron-beam) axis anddr <0.5cm in the transverser-ϕ plane. For a charged track, information from the detector subsystems is combined to form a likelihoodLifor a particle species ofi∈fe;μ;π; Kor protong[39]. Tracks withRK¼ LK=ðLKþLπÞ<0.4 are identified as pions with an effi- ciency of about 95%, while 6% of kaons misidentified as pions. Similar likelihood ratios are formed for electron and muon identification [40,41]. Both lepton candidates are required to haveRe >0.1 for theJ=ψ →eþe− mode; at least one candidate is required to have Rμ>0.1 for the J=ψ →μþμ−mode. For the first mode, any bremsstrahlung
photons detected in the ECL within 0.05 radians of the original lepton direction are included in the calculation of the eþe− invariant mass.
The invariant mass distributions of the lepton pair (Mlþl−) from data are shown in Fig. 1(a), where clear J=ψ signals are seen. By fitting the Mlþl− distributions with a Gaussian function for the J=ψ signal and a first- order polynomial function for background, we obtain the J=ψmass resolutions of11.00.6MeV=c2from data and 9.40.1MeV=c2 from signal MC simulation. A lepton pair is regarded as a J=ψ candidate if jMlþl−− mJ=ψj<4σJ=ψ, whereσJ=ψ≡11.0MeV=c2is taken from the resolution in data andmJ=ψ is the nominal mass ofJ=ψ [1]. Figure 1(b) shows the distributions of Mπþπ−J=ψ≡ Mπþπ−lþl− −Mlþl−þmJ=ψ from data, whereMπþπ−lþl− is the invariant mass of πþπ−lþl−. Fitting the Mπþπ−J=ψ
distributions with a Gaussian function forψð2SÞsignal and a first-order polynomial function for the background, we obtain theψð2SÞmass resolutions of2.800.21MeV=c2 from data and 2.520.04MeV=c2 from the signal MC simulation. The ψð2SÞ signal window is defined to be jMπþπ−J=ψ−mψð2SÞj<2.5σψð2SÞ, whereσψð2SÞ≡2.8MeV=c2 is taken from the resolution in data and mψð2SÞ is the nominal mass ofψð2SÞ[1]. To estimate the background in theψð2SÞreconstruction, the sideband regions are defined to be jMπþπ−J=ψ−mψð2SÞ9σψð2SÞj<3.75σψð2SÞ, which are 3 times the width of the signal region.
The background is dominated by eþe− →ψð2SÞ via ISR, whereψð2SÞis combined with a fake photon. Figure2 shows the distributions of the recoil mass squared M2recðγψð2SÞÞofγψð2SÞ. For two-photon collision events, there may be an outgoingeþe−pair traveling back-to-back along theeþe−beams so thatM2recðγψð2SÞÞ, corresponding to the mass squared of the outgoingeþe− pair, tends to be large. For ISR events, the recoil ofγψð2SÞis dominated by one energetic ISR photon with EðγISRÞ>1.5GeV, so M2recðγψð2SÞÞ is around zero. We apply M2recðγψð2SÞÞ>
10ðGeV=c2Þ2 to remove most ISR events. Nevertheless, there still remain events with two ISR photons traveling
back-to-back along theeþe− collision beams; such events have a topology similar to two-photon collisions.
To suppress the ISR background further, the transverse momenta of ψð2SÞ and γψð2SÞ, i.e., Ptðψð2SÞÞ and Ptðγψð2SÞÞ, calculated in the c.m. system and shown in Figs.3(a)and3(c), are used.Ptðγψð2SÞÞis small for most of the signal events, in which the outgoing eþe− travel along the accelerator beamline. However,Ptðψð2SÞÞcould be large ifψð2SÞoriginates from the decay of a resonance such asχc0ð2PÞorχc2ð2PÞ. For the ISR events,Ptðψð2SÞÞ is small since the ISR photon(s) always travel along the accelerator beamline. We optimize the selections of Ptðψð2SÞÞ and Ptðγψð2SÞÞ based on the Punzi figure of merit (FOM), defined as
FOM≡ εðtÞ
a=2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
NbkgðtÞ
p ð1Þ
according to Eq. (7) of Ref.[42]. Here,εðtÞis the signal efficiency based on the selection criteriont,ais the number
0 20 40 60 80
3 3.1 3.2
M(l+l-)
Entries/6 MeV/c2
GeV/c2
0 25 50 75 100
3.64 3.66 3.68 3.7 3.72 M(π+π-J/ψ)
Entries/2 MeV/c2
GeV/c2
(a) (b)
FIG. 1. Invariant mass distributions of (a)lþl−ðl¼e;μÞfor theJ=ψsignal and (b)πþπ−J=ψfor theψð2SÞsignal in data. The curves show the best fit results. The red solid arrows show the signal regions and the green dashed arrows show the sideband regions.
1 10 102
0 20 40
M2rec(γψ(2S)) Entries/0.6 GeV2 /c4
GeV2/c4
FIG. 2. The distributions of recoil mass square ofγψð2SÞ. The dots with error bars are data, the black blank histogram is the background estimated from theψð2SÞmass sidebands, the blue histogram is the ISR MC simulation and the green histogram is the signal MC simulation. The normalization of ISR MC simu- lation is according to the regionM2recðγψð2SÞÞ<10ðGeV=c2Þ2, and the one of the signal MC simulation is arbitrary. The arrow shows the position ofM2recðγψð2SÞÞ ¼10ðGeV=c2Þ2.
of sigmas corresponding to one-side Gaussian tests—we takea¼5—andNbkgðtÞis the background estimated from the ISR events and theψð2SÞ mass sidebands. We do the optimization on Ptðψð2SÞÞ and Ptðγψð2SÞÞ individually and iterate the procedure until both selections are at their optimal values. The FOM andεðtÞversusPt selections are shown in Figs.3(b) and3(d). We applyPtðψð2SÞÞ>0.1 and Ptðγψð2SÞÞ<0.2GeV=c with selection efficiencies of εMCðtÞ ¼ ð97.10.3Þ% and εMCðtÞ ¼ ð67.80.7Þ%,
respectively. There are about 150 ISR events surviving these selection criteria with an efficiency of about 0.02%.
IV. INVARIANT MASS DISTRIBUTION OFγψð2SÞ AND TWO STRUCTURES Figure4shows the invariant mass distributions for data ofγψð2SÞ(Mγψð2SÞ) in theJ=ψ →eþe− andμþμ− modes [43]. The distributions of the backgrounds estimated from
1 10 102
0 0.1 0.2 0.3 0.4 0.5
Pt*(ψ(2S))
Entries/5 MeV/c
GeV/c
0.01 0.012 0.014 0.016
0 0.05 0.1 0.15 0.2
Pt*(ψ(2S))
FOM
GeV/c 0.08 0.1 0.12
ε(t)
0 10 20 30 40
0 0.25 0.5 0.75 1
Pt*(γψ(2S))
Entries/10 MeV/c
GeV/c
0.01 0.012 0.014 0.016
0 0.1 0.2 0.3
Pt*(γψ(2S))
FOM
GeV/c 0.06 0.08 0.1 0.12
0 0.1 0.2 0.3
ε(t)
(a)
(b)
(c)
(d)
FIG. 3. The distributions of transverse momenta ofψð2SÞ(top row) andγψð2SÞ(bottom row) in the c.m. system ofeþe−collision. In the left panel, the dots with error bars are from the signal region, the shaded histograms are the backgrounds estimated from theψð2SÞ mass sidebands, the black blank histograms are signal MC simulation with arbitrary normalizations, and the blue blank histograms are ISR MC simulation. The normalization of the ISR MC simulation in (a) is according to data withPtðψð2SÞÞ<0.1GeV=c2, and the one in (c) is according to the size of the ISR MC simulation sample. The right panel shows the Punzi FOMs in dots and efficiencies in diamonds versus (b) Ptðψð2SÞÞand (d)Ptðγψð2SÞÞ. The arrows show the selections applied.
0 2.5 5 7.5 10
3.7 3.8 3.9 4 4.1 4.2
M(γψ(2S))
Entries/5 MeV/c2
GeV/c2
0 5 10 15 20
3.7 3.8 3.9 4 4.1 4.2
M(γψ(2S))
Entries/5 MeV/c2
GeV/c2
(a) (b)
FIG. 4. The invariant mass distributions ofγψð2SÞfrom (a)eþe−mode and (b)μþμ−mode. The dots with error bars are data, and the shaded histograms are backgrounds estimated from theψð2SÞmass sidebands. The blank histograms are ISR events simulated and scaled to the size of the Belle data sample.
the scaledψð2SÞmass sidebands and ISR events simulated by PHOKHARA [37] are also shown in Fig. 4. The ratio between data and ISR MC simulation is 0.1470.012 from the distributions in the regionMγψð2SÞ<3.9GeV=c2, while the expected ratio is0.1560.009according to the cross section and the size of the ISR MC sample. Figure5 shows the signal selection efficiency and the two-photon luminosity function Lγγð ffiffiffi
ps
Þ, which is defined as the probability of a two-photon emission withγγ c.m. system energy ffiffiffi
ps
in the Belle experiment[35]. The efficiencies for JPC¼0þþ and 2þþ (jλj ¼0 or 2) range from ∼10% to
∼15%forMγψð2SÞ between 3.85 and 4.20GeV=c2. The Mγψð2SÞ distribution of data after combining the eþe−andμþμ−modes is shown in Fig.6. Excesses around 3.92 and 4.02GeV=c2 are seen. Both χc2ð3930Þ and
Xð3915Þ have the mass close to 3.92GeV=c2, but no resonance with a mass close 4.02GeV=c2 has been discovered in prior experiments. To study the excesses, a binned extended maximum-likelihood fit is performed to theMγψð2SÞ mass spectra. The function used for the fit is characterized by the sum
fsum ¼fR1þfR2þfISRþfbkgþfSB: ð2Þ Here, fR1 (fR2) is for the structure R1 (R2) near 3.92GeV=c2 (4.02GeV=c2), fISR for the ISR events, fSB for the background in ψð2SÞ reconstruction, and fbkg for the possible additional backgrounds. The fSB
distribution is estimated from theψð2SÞ mass sidebands, and its yield is fixed in the fits. Assuming the orbital angular momentum is zero between γ and ψð2SÞ, the function fR1 (fR2) contains the convolution of a relati- vistic Breit-Wigner (BW) function with a form of 12πΓγγΓγψð2SÞ=ððs−M2Þ2þM2Γ2Þ and a Crystal Ball (CB) function [44] with a mass resolution of about 7.4MeV=c2 (8.1MeV=c2), and the parameters of CB function are fixed according to the signal MC simulation of a resonance with a mass near that of theR1(R2) state and with zero width. The resonant parameters in the BW function, viz. M, Γ, and Γγψð2SÞ (Γγγ), are the mass, the width, and the partial width of the decay to the final state γψð2SÞ(γγ), respectively. The productΓγψð2SÞΓγγis treated as one parameter, since it is impossible to separate Γγψð2SÞandΓγγ in the fits. The efficiency curveεis shown in Fig. 5(a) and is incorporated into fR1 and fR2, i.e., fR ∝εðBW⊗CBÞ. The widthsΓR1 andΓR2 are found to be small and thus the possible interference between R1 andR2 is expected to be small and is ignored in the fit.
The histogram of Mγψð2SÞ distribution from the ISR MC simulation is used forfISR. There may be more subdomi- nant sources of background, such as high order QED processes and continuum production ofγγ→γψð2SÞ, but their individual and collective contributions are not clearly
0.08 0.1 0.12 0.14 0.16
3.8 3.9 4 4.1 4.2
M(γψ(2S))
ε
GeV/c2
2 4 6 8
3.8 4 4.2
√s⎯ Lγγ/(0.001/GeV)
GeV (a)
(b)
FIG. 5. (a) The efficiencies at differentMγψð2SÞfrom MC simulation, and (b) the two-photon luminosity functionLγγð ffiffiffi ps
Þ. In (a), the blue dots are the efficiencies forJPC¼0þþ, and the black (red) dots are the efficiencies for2þþ with helicityjλj ¼0(jλj ¼2). The curves show the best fits with polynomial functions.
3.7 3.8 3.9 4 4.1 4.2
(2S)) GeV/c 2
ψ γ M(
0 5 10 15 20
2Events/5 MeV/c
BELLE
FIG. 6. The γψð2SÞ invariant mass distribution and the fit result. The points with error bars show the data while the shaded histogram is the normalized background from the ψð2SÞ mass sidebands. The solid blue curve shows the best fit results. The red signal curves from the convolutions of BW and CB functions show the contributions from the two structures. The green blank histogram shows the component of ISR events of eþe−→ψð2SÞ→πþπ−J=ψ. The pink dashed line shows the possible additional background, modeled by a second-order polynomial.
distinguishable with the current limited statistics. A second- order polynomial function is used forfbkg, and polynomial functions with different order are considered to estimate the systematic uncertainty.
The result from a fit in which all parameters are floated except the yield of thefSB component is shown Fig.6and Table I. The reduced chi squared of the fit to theMγψð2SÞ spectrum is χ2=ndf¼0.69. The signal yields are NR1¼ 3111events forR1 withMR1 ¼3922.46.5MeV=c2 andΓR1 ¼2217MeV, andNR2 ¼197events forR2
withMR2¼4014.34.0MeV=c2andΓR2 ¼411MeV.
The production ofR1andR2in two-photon collisions is studied by determining the parameterB·Γγγ≡Γγψð2SÞΓγγ=Γ with the formula[35]
B·Γγγ ¼ nsigfit Ltot·Bprod·ε·Fð ffiffiffi
ps
; JÞ; ð3Þ wherensigfit is the signal yield from the fit,Ltot ¼980 fb−1is the integrated luminosity of the Belle data sample,Jis the spin of a structure, and Bprod is the product of branching fractions Bðψð2SÞ→πþπ−J=ψÞ·BðJ=ψ →eþe−=μþμ−Þ.
Since ΓR1 and ΓR2 are small compared to the available kinetic energy in the decays, the spin-dependent factor is Fð ffiffiffi
ps
; JÞ ¼4π2ð2Jþ1ÞLγγð ffiffiffi ps
Þ=s. The best fit gives ΓγγBðR1→γψð2SÞÞ ¼ ð9.83.6ÞeV ifJ¼0andð2.0 0.7ÞeV if J¼2 for structure R1, and ΓγγBðR2→ γψð2SÞÞ ¼ ð6.82.8ÞeV if J¼0and ð1.40.6ÞeV if
J¼2 for structure R2. The ISR yield of 13415 is consistent with the estimate from the ISR MC simulation of 15410. The mass of R1 indicates that it is a good candidate for Xð3915Þ, χc2ð3930Þ or an admixture of them. An alternate fit with both structures included and MR1andΓR1fixed to the nominalXð3915Þparameters yields ΓγγBðXð3915Þ→γψð2SÞÞ ¼9.62.91.1eV ifJPC¼ 0þþ and1.90.60.2eV ifJPC¼2þþ. Another alter- native fit with the mass and width ofR1fixed to those of χc2ð3930ÞyieldsΓγγBðχc2ð3930Þ→γψð2SÞÞ ¼2.20.6 0.4eV ifJPC¼2þþ. A third alternate fit withR1being an admixture of Xð3915Þ and χc2ð3930Þ shows no notable change in the fit quality. The systematic uncertainties here are described in Sec.V.
The local signal significance is determined to be3.5σfor R1and3.4σforR2by comparing the value ofΔð−2lnLÞ ¼
−2lnðLmax=L0Þ and the change of the number of free parameters (Npar) in the fits, whereLmax is the likelihood with both R1 and R2 included in Eq. (2), and L0 is the likelihood with only one ofR1orR2excluded. The values of−2lnL,χ2=ndf, andNparof these fits are summarized in Table II. The local signal significance of R1 is deter- mined to be4.1σ(3.9σ) in the case that its mass and width are fixed to those of Xð3915Þ (χc2ð3930Þ). Taking into account the systematic uncertainties, described in Sec.V, the lowest value of the local significance of R1 is 3.1σ. SinceR2 has never been seen before, the look-elsewhere effect is assessed for it with pseudo-experiments to check its global significance. The function for generating pseudo- experiments isftoyMC¼fR1þfISRþfbkgþfSBwith the parameters from the nominal fit. The fit in each pseudo- experiment is performed with the same procedures as for the nominal fit to the actual data sample, except that the mass range of R2 is limited to MR2 >3.95GeV=c2 because the regionMγψð2SÞ<3.95GeV=c2 is dominated byR1and ISR backgrounds. Among the5.0×104pseudo- experiments, the number of experiments withΔð−2lnLÞ ofR2signal larger than the one from data is 137. Therefore, the probability considering the look-elsewhere effect is about ð2.740.23Þ×10−3, corresponding to a global significance of2.8σ. Since the mass ofR1is close to that of Xð3915Þ or χc2ð3930Þ and the width—with its large uncertainty—has no conflict with that of Xð3915Þ or TABLE I. Summary of the resonant parameters determined.
The units of mass (M), width (Γ), product of partial width and branching fractionΓγγBare MeV=c2, MeV and eV, respectively.
The first errors are statistical and the second are systematic.
Resonant parameters J¼0 J¼2
MR1 3922.46.52.0
ΓR1 22174
ΓγγBðR1→γψð2SÞÞ 9.83.61.3 2.00.70.2 MR2 4014.34.01.5
ΓR2 4116
ΓγγBðR2→γψð2SÞÞ 6.22.20.8 1.20.40.2
TABLE II. The values of−2lnL,χ2=ndfand number of free parameters (Npar) in the different fits. From left to right, the rows are the fits with no resonance included, onlyR1included, onlyR2included, bothR1andR2included (nominal fit), both resonances included and the mass and width ofR1fixed to those ofXð3915Þ, and both resonances included and the mass and width of R1 fixed to those of χc2ð3930Þ. Only the differences among−2lnL are meaningful in studying the statistical significance ofR1and R2.
… No resonance R1 only R2 only R1þR2 Xð3915Þ þR2 χc2ð3930Þ þR2
−2lnL −2932.2 −2946.5 −2946.3 −2965.4 −2964.8 −2963.0
χ2=ndf 0.76 0.74 0.78 0.70 0.68 0.68
Npar 5 8 8 11 9 9
χc2ð3930Þ, we do not treatR1as a new never-observed state and so the look-elsewhere effect study is not performed to it.
V. SYSTEMATIC UNCERTAINTIES
There are systematic uncertainties in determining the resonant parameters of the two structures. The masses and widths are determined from fitting to the invariant mass distribution ofγψð2SÞ. In determiningB·Γγγwith Eq.(3), additional systematic uncertainties from the selection efficiency, the luminosity of Belle data sample and the branching fractions ofJ=ψandψð2SÞdecays are taken into account.
The uncertainties due to the fits are estimated by changing the fit range, the number and the fSB shape of the background in theψð2SÞreconstruction, thefbkgshape, the bin width of theMγψð2SÞ distribution, the parametriza- tion of the BW function, and the resonant parameters of Xð3915Þ and χc2ð3930Þ. The fit range is changed from [3.70, 4.20] to½3.725;4.15 GeV=c2. The number ofψð2SÞ mass sideband events is changed by 1σ, and the sideband region is changed from jMπþπ−J=ψ−mψð2SÞ9σψð2SÞj<
3.75σψð2SÞ to jMπþπ−J=ψ −mψð2SÞ8σψð2SÞj<3.75σψð2SÞ to estimate the uncertainty due to the fSB component.
Another ISR MC sample is simulated to estimate the uncertainty from the shape of fISR. The alternative poly- nomial function forfbkgis first order or third order. The bin width is changed from 5 to 4MeV=c2. The alternative formula of the resonant shape is BW∝ðM2=sÞ· 12πΓγγΓX=ððs−M2Þ2þM2Γ2Þ. The uncertainty from the resolution of Mγψð2SÞ is mainly related to the recon- structed γ, and it is estimated with a sample of about 4;000γγ →χc2→γJ=ψ events selected in the Belle data sample. Fitting to χc2 signals in the MγJ=ψ distributions from data and MC simulation results in the consistent value of 10.840.26 and 10.770.22MeV=c2, respectively.
Thus, the uncertainty due to the mass resolution ofγψð2SÞ is expected to be very small and so is ignored. WhenMR1
andΓR1 are fixed to those of Xð3915Þor χc2ð3930Þ, their values are changed by1σto estimate the related systematic uncertainties [1]. The largest differences between the nominal fit results and those from these various fits are taken as the systematic uncertainties of the mass, the width and the productΓγγBðR→γψð2SÞÞ. A fit bias study using 200 toy MC samples shows that the bias ofNR1 is less than 3%, and those of other parameters are negligible. The statistics of each toy MC sample is 500 times of data to avoid the large fluctuations in testing the fit procedure. We take 3% to be the systematic uncertainty due to the fit bias for NR1.
Several sources of non-fit-related systematic uncertain- ties are considered. The particle identification uncertainty is 2.8% [39–41]; the uncertainty of the tracking efficiency is 0.35% per track and is additive; the uncertainty of the
photon reconstruction is 2% per photon. The efficiency for the tracks in the extreme forward and backward regions obtained from MC simulation is found to be higher than that obtained in data according to the study of eþe−→ψð2SÞ→πþπ−J=ψ via ISR [36], and appro- priate corrections have been applied. The uncertainty in the ψð2SÞ mass window requirement is measured to be 0.6%, while the one of theJ=ψ mass window is ignored.
The efficiencies of the selection criteria on Ptðψð2SÞÞ and Ptðγψð2SÞÞ are strongly related to the boost trans- formation from the lab system to the c.m. system ofeþe− collisions. However, the related uncertainty is very small, and 1% is taken to be a conservative estimation for the uncertainty due to the Pt selections. The uncertainty due to the M2recðγψð2SÞÞ requirement is less than 0.5%.
The uncertainty due to the momentum and angular distributions of helicities 0 and 2 for J¼2 is estimated to be 4.3% from the TREPSgenerator[35], while the one of J¼0 is ignored with the decay to γψð2SÞ isotropic and no uncertainty in helicity. The systematic uncertainty of the luminosity function from TREPS is 2.5%, which includes 1.1% from the calculation, under 1.0% from the form factor and 1%–2% from the radiative-correction effect [45]. Belle measures the luminosity with 1.4%
precision. The trigger efficiency for the events surviving the selection criteria exceeds 99.4%, and so the uncer- tainty is ignored. The uncertainties of theJ=ψ andψð2SÞ decay branching fractions taken from Ref. [1] con- tribute a systematic uncertainty of 1.3%. The statistical error in the MC determination of the efficiency is less than 0.7%.
The non-fit-related systematic uncertainties are listed in TableIII. Assuming all the sources are independent, we add them in quadrature to obtain a total systematic uncertainty of 6.6% (5.1%) ofJ¼2(J¼0) in determiningB·Γγγ, in addition to the uncertainties from the fits.
TABLE III. The summary of systematic uncertainties besides the fits inγγ→γψð2SÞmeasurement.
Source Relative error (%)
… J¼0 J¼2
Particle identification 2.8
Tracking efficiency 1.4
Photon reconstruction 2.0
ψð2SÞmass window 0.6
Ptðψð2SÞÞand Ptðγψð2SÞÞ 1.0
M2recðγψð2SÞÞ 0.5
Integrated luminosity 1.4
Helicity … 4.3
Luminosity function 2.5
Branching fractions 1.3
Statistics of MC samples 0.7
Sum in quadrature 5.1 6.6
VI. DISCUSSION ON THE TWO STRUCTURES We find evidence for the structureR1near3.92GeV=c2, which may be Xð3915Þ, χc2ð3930Þ, or an admixture of them. Assuming R1 is χc2ð3930Þand taking into account ΓγγBðχc2ð3930Þ→DD¯Þ ¼21040eV, the ratio R¼ Bðχc2ð3930Þ→γψð2SÞÞ=Bðχc2ð3930Þ→DDÞ ¼¯ 0.010 0.003 is obtained. A rough estimation shows the partial widthΓðχc2ð3930Þ→γψð2SÞÞ ¼ ð200∼300ÞkeV, which is close to the predicted value of 207 keV from the Godfrey-Isgur relativistic potential model [30].
It is interesting to see that the mass ofR2agrees with the HQSS-predicted mass (≈4013MeV=c2) of the2þþpartner of Xð3872Þ [18]. The mass difference between R2 and Xð3872Þ is 142.64.2MeV=c2, while that between D0ð2007Þ and D0 is 142.01MeV=c2. Meanwhile, the width ofR2from the fit coincides with the predicted width of2–8 MeV=c2for the2þþpartner ofXð3872Þ[19]. Thus, R2 may provide important information for understanding the nature of theXð3872Þ. However, the global significance ofR2is only2.8σ. A much larger data sample that will be collected by Belle II may resolve this in the near future.
VII. SUMMARY
The two-photon processγγ →γψð2SÞis studied in the γγ mass range from the threshold to 4.2GeV=c2 for the first time with the full Belle data sample, and two structures are seen in the invariant mass distribution ofγψð2SÞ. The first has a mass ofMR1 ¼3922.46.52.0 MeV=c2and a width ofΓR1 ¼22174MeV with a local statistical significance of3.1σwhen the systematic uncertainties are included. This is close to the mass of Xð3915Þ and χc2ð3930Þ. The second has a mass of MR2 ¼4014.3 4.01.5MeV=c2and a width ofΓR2 ¼4116 MeV, with a global statistical significance of2.8σ. The values of ΓγγBðR→γψð2SÞÞ are of the order of several eV.
ACKNOWLEDGMENTS
We thank the KEKB group for the excellent operation of the accelerator; the KEK cryogenics group for the efficient operation of the solenoid; and the KEK computer group, and the Pacific Northwest National Laboratory (PNNL) Environmental Molecular Sciences Laboratory (EMSL) computing group for strong computing support; and the National Institute of Informatics, and Science Information NETwork 5 (SINET5) for valuable network support. We acknowledge support from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of
Japan, the Japan Society for the Promotion of Science (JSPS), and the Tau-Lepton Physics Research Center of Nagoya University; the Australian Research Council including Grants No. DP180102629, No. DP170102389, No. DP170102204, No. DP150103061, and No. FT130100303; Austrian Federal Ministry of Education, Science and Research (FWF) and FWF Austrian Science Fund No. P 31361-N36; the National Natural Science Foundation of China under Contracts No. 11435013, No. 11475187, No. 11521505, No. 11575017, No. 11675166, No. 11705209, and No. 12175041; Key Research Program of Frontier Sciences, Chinese Academy of Sciences (CAS), Grant No. QYZDJ-SSW-SLH011; the CAS Center for Excellence in Particle Physics (CCEPP); the Shanghai Science and Technology Committee (STCSM) under Grant No. 19ZR1403000; the Ministry of Education, Youth and Sports of the Czech Republic under Contract No. LTT17020; Horizon 2020 ERC Advanced Grant No. 884719 and ERC Starting Grant No. 947006
“InterLeptons” (European Union); the Carl Zeiss Foundation, the Deutsche Forschungsgemeinschaft, the Excellence Cluster Universe, and the Volkswagen- Stiftung; the Department of Atomic Energy (Project Identification No. RTI 4002) and the Department of Science and Technology of India; the Istituto Nazionale di Fisica Nucleare of Italy; National Research Foundation (NRF) of Korea Grants No. 2016R1D1A1B01010135, No. 2016R1D1A1B02012900, No. 2018R1A2B3003643, No. 2018R1A6A1A06024970, No. 2018R1D1A1B- 07047294, No. 2019K1A3A7A09033840, and No. 2019R1I1A3A01058933; Radiation Science Research Institute, Foreign Large-size Research Facility Application Supporting project, the Global Science Experimental Data Hub Center of the Korea Institute of Science and Technology Information and KREONET/GLORIAD; the Polish Ministry of Science and Higher Education and the National Science Center; the Ministry of Science and Higher Education of the Russian Federation, Agreement No. 14.W03.31.0026, and the HSE University Basic Research Program, Moscow; University of Tabuk research Grants S-1440-0321, S-0256-1438 and No. S-0280-1439 (Saudi Arabia); the Slovenian Research Agency Grants No. J1-9124 and No. P1-0135; Ikerbasque, Basque Foundation for Science, Spain; the Swiss National Sci- ence Foundation; the Ministry of Education and the Ministry of Science and Technology of Taiwan; and the United States Department of Energy and the National Science Foundation.
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